124 research outputs found
The extremal spectral radii of -uniform supertrees
In this paper, we study some extremal problems of three kinds of spectral
radii of -uniform hypergraphs (the adjacency spectral radius, the signless
Laplacian spectral radius and the incidence -spectral radius).
We call a connected and acyclic -uniform hypergraph a supertree. We
introduce the operation of "moving edges" for hypergraphs, together with the
two special cases of this operation: the edge-releasing operation and the total
grafting operation. By studying the perturbation of these kinds of spectral
radii of hypergraphs under these operations, we prove that for all these three
kinds of spectral radii, the hyperstar attains uniquely the
maximum spectral radius among all -uniform supertrees on vertices. We
also determine the unique -uniform supertree on vertices with the second
largest spectral radius (for these three kinds of spectral radii). We also
prove that for all these three kinds of spectral radii, the loose path
attains uniquely the minimum spectral radius among all
-th power hypertrees of vertices. Some bounds on the incidence
-spectral radius are given. The relation between the incidence -spectral
radius and the spectral radius of the matrix product of the incidence matrix
and its transpose is discussed
Connected Hypergraphs with Small Spectral Radius
In 1970 Smith classified all connected graphs with the spectral radius at
most . Here the spectral radius of a graph is the largest eigenvalue of its
adjacency matrix. Recently, the definition of spectral radius has been extended
to -uniform hypergraphs. In this paper, we generalize the Smith's theorem to
-uniform hypergraphs. We show that the smallest limit point of the spectral
radii of connected -uniform hypergraphs is . We
discovered a novel method for computing the spectral radius of hypergraphs, and
classified all connected -uniform hypergraphs with spectral radius at most
.Comment: 20 pages, fixed a missing class in theorem 2 and other small typo
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